5478 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 10, OCTOBER 2012

Sum-Rate Maximization in Two-Way AF MIMORelaying: Polynomial Time Solutions to a Class of

DC Programming ProblemsArash Khabbazibasmenj, Student Member, IEEE, Florian Roemer, Student Member, IEEE,Sergiy A. Vorobyov, Senior Member, IEEE, and Martin Haardt, Senior Member, IEEE

Abstract—Sum-rate maximization in two-way amplify-and-forward (AF) multiple-input multiple-output (MIMO) relayingbelongs to the class of difference-of-convex functions (DC) pro-gramming problems. DC programming problems occur alsoin other signal processing applications and are typically solvedusing different modifications of the branch-and-bound methodwhich, however, does not have any polynomial time complexityguarantees. In this paper, we develop two efficient polynomialtime algorithms for the sum-rate maximization in two-way AFMIMO relaying. The first algorithm guarantees to find at least aKarush-Kuhn-Tucker (KKT) solution. There is a strong evidence,however, that such a solution is actually globally optimal. Thesecond algorithm that is based on the generalized eigenvectorsshows the same performance as the first one with reduced compu-tational complexity.The objective function of the problem is represented as a product

of quadratic fractional ratios and parameterized so that its convexpart (versus the concave part) contains only one (or two) optimiza-tion variables. One of the algorithms is called POlynomial TimeDC (POTDC) and is based on semi-definite programming (SDP)relaxation, linearization, and an iterative Newton-type search overa single parameter. The other algorithm is called RAte-maximiza-tion via Generalized EigenvectorS (RAGES) and is based on thegeneralized eigenvectors method and an iterative search over two(or one, in its approximate version) optimization variables. We de-rive an upper-bound for the optimal value of the corresponding op-

Manuscript received November 25, 2011; revised May 11, 2012; acceptedJune 23, 2012. Date of publication July 13, 2012; date of current versionSeptember 11, 2012. The associate editor coordinating the review of this paperand approving it for publication was Dr. Francesco Verde. This work has beensupported in part by the Natural Science and Engineering Research Council(NSERC) of Canada, by the Carl Zeiss Award, Germany, and by the GraduateSchool on Mobile Communications (GSMobicom), Ilmenau University ofTechnology, which is partly funded by the Deutsche Forschungsgemeinschaft(DFG). A part of this work has been performed in the framework of theEuropean research project SAPHYRE, which is partly funded by the EuropeanUnion under its FP7 ICT Objective 1.1—The Network of the Future. Parts ofthis paper have been presented at the European Signal Processing Conference(EUSIPCO), Aalborg, Denmark, 2010, and the IEEE International Conferenceon Acoustics, Speech and Signal Processing (ICASSP), Kyoto, Japan, 2012.A. Khabbazibasmenj and S. A. Vorobyov are with the Department of Elec-

trical and Computer Engineering, University of Alberta, Edmonton, AB T6G2V4, Canada (e-mail: khabbazi@ualberta.ca; svorobyo@ualberta.ca).F. Roemer and M. Haardt are with the Communication Research Laboratory,

Ilmenau University of Technology, Ilmenau, 98693, Germany (e-mail: florian.roemer@tu-ilmenau.de; martin.haardt@tu-ilmenau.de).This paper has supplementary downloadable multimedia material available

at http://ieeexplore.ieee.org provided by the authors. This includes the file“POTDCsumrate.rar” containing Matlab codes needed to generate the simu-lation results shown in the paper. This material is 20 KB in size and containsalso Readme.txt file.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2012.2208635

timization problem and show by simulations that this upper-boundis achieved by both algorithms. It provides an evidence that the al-gorithms find a global optimum. The proposed methods are alsosuperior to other state-of-the-art algorithms.

Index Terms—Difference-of-convex functions (DC) program-ming, non-convex programming, semi-definite programmingrelaxation, sum-rate maximization, two-way relaying.

I. INTRODUCTION

T WO-WAY relaying has recently attracted a significant re-search interest due to its ability to overcome the draw-back of conventional one-way relaying, that is, the factor of1/2 loss in the rate [1], [2]. Moreover, two-way relaying canbe viewed as a certain form of network coding [3] which al-lows to reduce the number of time slots used for the transmis-sion in one-way relaying by relaxing the requirement of ‘or-thogonal/non-interfering’ transmissions between the terminalsand the relay [4]. Specifically, simultaneous transmissions bythe terminals to the relay on the same frequencies are allowed inthe first time slot, while a combined signal is broadcasted by therelay in the second time slot. In contrast to the one-way relaying,the rate-optimal strategy for two-way relaying is in general un-known [5]. However, some efficient strategies have been devel-oped. Depending on the ability of the relay to regenerate/de-code the signals from the terminals, several two-way transmis-sion protocols have been introduced and studied. The regener-ative relay adopts the decode-and-forward (DF) protocol andperforms the decoding process at the relay [6], while the non-re-generative relay typically adopts a form of amplify-and-forward(AF) protocol and does not perform decoding at the relay, butamplifies and possibly beamforms or precodes the signals to re-transmit them back to the terminals [5], [7], [8]. The advan-tages of the latter are a smaller delay in the transmission andlower hardware complexity of the relay. Most of the researchon two-way relaying systems concentrates on studying the cor-responding sum-rate, the achievable rate region, and also the biterror probability of different schemes [9]. The tradeoff betweenthe error probability and the achievable rate has been recentlystudied in [9] using Gallager’s random coding error exponent.In this paper, we consider the AF two-way relaying system

with two terminals equipped with a single antenna and one relaywith multiple antennas. The task is to find the relay transmitstrategy that maximizes the sum-rate of both terminals. This isa basic model which can be extended in many ways. The sig-

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KHABBAZIBASMENJ et al.: SUM-RATE MAXIMIZATION IN TWO-WAY AF MIMO RELAYING 5479

nificant advantage of considering this basic model is that thecorresponding capacity region is discussed in the existing liter-ature [4]. It enables us to concentrate on the mathematical issuesof the corresponding optimization problem which are of signif-icant and ubiquitous interest.We show that the optimization problem of finding the relay

amplification matrix for the considered AF two-way relayingsystem is equivalent to finding the maximum of the productof quadratic fractional functions under a quadratic powerconstraint on the available power at the relay. Such a problembelongs to the class of the so-called difference-of-convexfunctions (DC) programming problems. It is worth stressingthat DC programming problems are very common in signalprocessing and, in particular, signal processing for communi-cations. For example, the robust adaptive beamforming for thegeneral-rank (distributed source) signal model with a positivesemi-definite constraint can be shown to belong to the classof DC programming problems [10], [11]. Specifically, theconstraint in the corresponding optimization problem is the dif-ference of two weighted norm functions. The power control forwireless cellular systems is also a DC programming problemwhen the rate is used as a utility function [12]. Similarly, thedynamic spectrum management for digital subscriber lines [13]as well as the problems of finding the weighted sum-rate point,the proportional-fairness operating point, and the max-minoptimal point (egalitarian solution) for the two-user mul-tiple-input single-output (MISO) interference channel [14]are all DC programming problems. The typical approach forsolving such problems is the use of various modifications ofthe branch-and-bound method [14]–[20] that is an efficientglobal optimization method. The branch-and-bound methodis known to work well especially for the case of monotonicfunctions, i.e., the case which is typically encountered in signalprocessing and, in particular, signal processing for communi-cations. However, it does not have any worst-case polynomialtime complexity guarantees, which significantly limits or evenprohibits its applicability in practical communication systems.Thus, methods with guaranteed polynomial time complexitythat can find at least a suboptimal solution for different types ofDC programming problems are of a great importance.In the last decade, a significant progress has occurred in

the application of optimization theory in signal processingand communications. Some of those results are relevant forthe considered problem of maximizing constrained product ofquadratic fractional functions [21]–[25]. The worst-case-basedrobust adaptive beamforming problem is known to belong tothe class of second-order cone (SOC) programming problems[21] largely due to the fact that the output signal-plus-in-terference-to-noise ratio (SINR) of adaptive beamformingis unchanged when the beamforming vector undergoes anarbitrary phase rotation. This allows to simplify the singleworst-case distortionless response constraint of the optimiza-tion problem into the form of a SOC constraint. The situationis more complicated in the case of multiple constraints of thesame type as the constraint in [21] when a single rotation ofthe beamforming vector is no longer sufficient to satisfy allconstraints simultaneously. This situation has been successfully

addressed in [22] by considering the semi-definite programming(SDP) relaxation technique. The SDP relaxation technique hasbeen further developed and studied in, for example, [23]–[25]and other works. Interestingly, the work [25] considers thefractional quadratically constrained quadratic programming(QCQP) problem that is closest mathematically to the one ad-dressed in this paper with the significant difference though thatthe objective in [25] contains only a single quadratic fractionalfunction that simplifies the problem dramatically.In this paper, we develop two polynomial time algorithms

for solving at least sub-optimally the non-convex DC program-ming problem of maximizing a product of quadratic fractionalfunctions under a quadratic constraint, which precisely corre-sponds to the sum-rate maximization in two-way AFMIMO re-laying.1 Our algorithms use such parameterizations of the ob-jective function that its convex part (versus the concave part)contains only one (or two) optimization variables. One of the al-gorithms is named POlynomial Time DC (POTDC) and is basedon SDP relaxation, linearization, and an iterative Newton-typesearch over a single parameter. It is guaranteed that it findsat least a Karush-Kuhn-Tucker (KKT) solution, i.e., a solutionwhich satisfies the KKT optimality conditions. The POTDC al-gorithm is rigorous and there is great evidence that the KKTsolution found by it is also globally optimal. Indeed, the so-lution given by POTDC coincides with the newly developedupper-bound for the optimal value of the problem. The otheralgorithm is called RAte-maximization via Generalized Eigen-vectorS (RAGES) and is based on the generalized eigenvectorsmethod and an iterative search over two (or one, in its approxi-mate version) optimization variables. The RAGES algorithm issomewhat heuristic in its approximate version, but may enjoy alower complexity. It shows, however, the same performance asthe first one.The rest of the paper is organized as follows. The two-wayAF

MIMO relaying system model is given in Section II while thesum-rate maximization problem for the corresponding system isformulated in Section III. The POTDC algorithm for the sum-rate maximization is developed in Section IV and an upper-bound for the optimal value of the maximization problem isfound in Section V. In Section VI, the RAGES algorithm isdeveloped and investigated. Simulation results are reported inSection VII. Finally, Section VIII presents our conclusions anddiscussions. This paper is reproducible research [28], and thesoftware needed to generate the simulation results can be ob-tained from the IEEE Xplore together with the paper.

II. SYSTEM MODEL

We consider a two-way relaying system with two single-an-tenna terminals and an AF relay equipped with antennas.Fig. 1 shows the system we study in the paper. In the first trans-mission phase, both terminals transmit to the relay. Assumingfrequency-flat quasi-static block fading, the received signal atthe relay can be expressed as

(1)1Some preliminary results have been presented in [26] and [27].

5480 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 10, OCTOBER 2012

Fig. 1. Two-way relaying system model.

where represents the (for-ward) channel vector between terminal and the relay, is thetransmitted symbol from terminal , denotes theadditive noise component at the relay, and stands for thetranspose of a vector or a matrix. Let be theaverage transmit power of terminal andbe the noise covariance matrix at the relay, where denotesthe mathematical expectation, and stands for the Hermitiantranspose of a vector or a matrix. For the special case of whitenoise we have where ,

is the identity matrix of size and denotesthe trace operation applied to a square matrix. The relay ampli-fies the received signal by multiplying it with a relay amplifica-tion matrix , i.e., it transmits the signal

(2)

The transmit power used by the relay can be expressed as

(3)

where denotes the Euclidean norm of a vector andis the covariance matrix of which is given by

(4)

The covariance matrix is assumed to be full rank whichis true under the common practical assumption that the noisecovariance matrix is full rank. However, the case of rankdeficient is considered for completeness in Appendix A aswell.Using the equality

(5)

which holds for any arbitrary square matrices and , the totaltransmit power of the relay (3) can be equivalently expressed as

(6)

where stands for the vectorization operation that trans-forms a matrix into a long vector stacking the columns of thematrix one after another. Finally, using the equality

, which is valid for any arbitrary squarematrices and , (6) can be equivalently rewritten as the fol-lowing quadratic form

(7)

where , and denotes the Kro-necker product. Since is the Kronecker product of the twofull rank positive definite matrices and , it is also fullrank and positive definite [29].In the second phase, the terminals receive the relay’s trans-

mission via the (backward) channels and (inthe special case when reciprocity holds we havefor , 2). Consequently, the received signals , , 2 atboth terminals can be expressed, respectively, as

(8)

(9)

where is the effective channel between

terminals and for , , 2 andrepresents the effective noise contribution at terminal whichcomprises the terminal’s own noise as well as the forwardedrelay noise. The first term in the received signal of each terminalrepresents the self-interference, which can be subtracted by theterminal since its own transmitted signal is known. The requiredchannel knowledge for this step can be easily obtained, for ex-ample, via the Least Squares (LS) compound channel estimatorof [30].After the cancellation of the self-interference, the two-way re-

laying system is decoupled into two parallel single-user single-input single-output (SISO) systems. Consequently, the rate ofterminal can be expressed as

(10)

where and are the powers of the desired signal andthe effective noise term at terminal , respectively, and

. Specifically, ,

, and for , 2. Note thatthe factor 1/2 results from the two time slots needed for the bidi-rectional transmission. The powers of the desired signal and theeffective noise term at terminal can be equivalently expressedas

(11)

(12)

(13)

where the expectation is taken with respect to the transmit sig-nals and also the additional noise terms, denotes the vari-ance of the additive noise at terminal , i.e., and standsfor the conjugation. Moreover, these powers can be further ex-pressed as quadratic forms in . For this goal, first note that byusing the following equality

(14)

KHABBAZIBASMENJ et al.: SUM-RATE MAXIMIZATION IN TWO-WAY AF MIMO RELAYING 5481

which is valid for any arbitrary matrices , and of com-patible dimensions, the term can be modifiedas follows

(15)

Using (15), the power of the desired signal at the first terminalcan be expressed as

(16)

Applying also the equalityto (16) which is valid for any arbitrary matrices , ,

and of compatible dimensions, can be expressed asthe following quadratic form

(17)The power of the desired signal at the second terminal, i.e., ,can be obtained similarly. By defining the matrices and

as follows

(18)

(19)

the powers of the desired signal can be expressed as

(20)(21)

Since thematrices , , 2 and ,, 2 are all positive semi-definite and the Kronecker product ofpositive semi-definite matrices is a positive semi-definite matrix[29], the matrices and are also positive semi-definite.As the last step, the effective noise can be converted intoa quadratic form of through the following train of equalities

(22)

(23)

(24)

(25)(26)

where (24) is obtained from (23) by applying the equality (5),(25) is obtained from (24) by applying the equality (14), and thematrix in (26) is defined as

(27)

Note that, , , 2 are positive semi-definite matricesbecause and , 1, 2 are positivesemi-definite.

III. PROBLEM STATEMENTOur goal is to find the relay amplification matrix which

maximizes the sum-rate subject to a power constraint atthe relay. For conveniencewe express the objective function andits solution in terms of . Then the power constrained sum-ratemaximization problem can be expressed as

(28)

where is the total available transmit power at the relay.Using the definitions from the previous section, this optimiza-tion problem can be rewritten as

(29)

(30)

where we have used the fact that is a monotonicfunction in , and , , 2 are defined after (10).It is worth noting that the inequality constraint in the opti-

mization problem (30) has to be active at the optimal point. Thiscan be easily shown by contradiction. Assume that satisfies

. Then we can find a constant suchthat satisfies . The latterfollows from the fact that is positive definite and, therefore,

is positive. However, inserting in the objec-tive function of (29), we obtain

(31)

which is monotonically increasing in . Since we have ,the vector provides a larger value of the objective func-tions than which contradicts the assumption that wasoptimal.As a result, we have shown that the optimal vector

must satisfy the total power constraint of the problem (30)with equality, i.e., . Using this fact,the inequality constraint in the problem (30) can be replacedby the constraint . This enables us to sub-stitute the constant term , which appears in the effectivenoise power at terminal (26), with the quadratic term of

. This leads to an equivalent homoge-

neous expression for the ratio of , 1, 2. Thus, by using

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such substitution, , 1, 2 from (26) can be equivalentlywritten as

(32)

where is defined as

(33)

Inserting (20), (21), and (33) into (30), the optimizationproblem becomes

(34)

where we have defined the new matricesand . Since the matrices ,

, 2, , and are positive semi-definite and is a fullrank positive definite matrix, the matrices , , , andare all full rank positive definite matrices and hence invertible.Moreover, , , , and are all matrices.As a final simplifying step we observe that the objective func-

tion of (34) is homogeneous in , meaning that an arbitraryrescaling of has no effect on the value of the objective func-tion. Consequently, the equality constraint can be dropped sinceany solution to the unconstrained problem can be rescaled tomeet the equality constraint without any loss in terms of the ob-jective function. Therefore, the final form of our problem state-ment is given by

(35)

The optimization problem (35) can be interpreted as theproduct of two Rayleigh quotients (quadratic fractional func-tions). Moreover, it can be expressed as a DC programmingproblem. Indeed, as we will show later the objective function ofthe problem (35) can be written as a summation of two concavefunctions with positive sign and one concave function withnegative sign. Thus, the objective of the equivalent problemis, in fact, the difference of convex functions which is ingeneral non-convex. The available in the literature algorithmsfor solving such DC programming problems are based onthe so-called branch-and-bound method that does not haveany polynomial time computational complexity guarantees[14]–[20]. However, as we show next, at least a KKT solutionof the problem (35) can be found in polynomial time with agreat evidence that such a solution is also globally optimal.

IV. POLYNOMIAL TIME ALGORITHM FOR THE SUM-RATEMAXIMIZATION PROBLEM IN TWO-WAY AF MIMO RELAYING

Since the problem (35) is homogeneous, without loss of gen-erality, we can fix the quadratic term to be equal toone at the optimal point. By doing so and also by defining theadditional variables and , the problem (35) can be equiva-lently recast as

(36)

For future reference, we need the range of the variable . Due tothe fact that the quadratic function is set to one, thisrange can be easily obtained. Specifically, the smallest value offor which the problem (36) is still feasible can be obtained by

solving the following problem

(37)

Note that does not impose any restriction on the smallest pos-sible value of , because if denotes the optimal solutionof the problem (37), then can be chosen as

. Since the matrix is positive definite, it can be decom-

posed as where is a square root ofand it is invertible due to the properties in [31]. By defining the

new vector , i.e., the problem(37) is equivalent to

(38)

It is well known that according to the minimax theorem [32],the optimal value of (38) is the smallest eigenvalue of the matrix

. Using the fact that for any arbitrary squarematrices and , the eigenvalues of the matrix productsand are the same [33], it can be concluded that the

smallest eigenvalue of is the same as the

smallest eigenvalue of or, equivalently,.

The largest value of for which the problem (36) is still fea-sible can be obtained in a similar way, and it is equal to thelargest eigenvalue of the matrix . As a result, the rangeof is where and

denote the smallest and the largest eigenvalue opera-tors, respectively. Note that, since the matrices and arepositive definite (hence is also positive definite), the eigen-values of the product are all positive due to the prop-erties in [31] including .For future reference, we also define the following function ofand

(39)

where is the set of all pairs such that thecorresponding optimization problem obtained fromfor fixed and is feasible. The function is calledan optimal value function. Therefore, using the optimal valuefunction , the original optimization problem (36) can beequivalently recast as

(40)

Introducing the matrix and observing that for anyarbitrary matrix , the relationship

KHABBAZIBASMENJ et al.: SUM-RATE MAXIMIZATION IN TWO-WAY AF MIMO RELAYING 5483

holds, the optimal value function can be equivalentlyrecast as [34]

(41)

where stands for the rank of a matrix. In the optimiza-tion problem obtained from the optimal value function(41) by fixing and , the rank-one constraintis the only non-convex constraint with respect to the new

optimization variable . Using the SDP relaxation, the corre-sponding optimization problem can be relaxed by dropping therank-one constraint, and the following new optimal value func-tion can be defined

(42)

where is the set of all pairs such that the opti-mization problem corresponding to for fixed and isfeasible. For brevity, we will refer to the optimization problemscorresponding to the functions and when andare fixed simply as the optimization problems of and

, respectively. The following lemma finds the relation-ship between the domains of the functions and .Lemma 1: The domains of the functions and

are the same, i.e., .Proof: First, we prove that if then .

Let . It implies that there exists a vector such thatthe constraints , , and

are satisfied. Defining the new matrix ,it is easy to verify that satisfies the constraints in the opti-mization problem of and, therefore, . Now,let us assume that . Therefore, there exists a positivesemi-definite matrix with rank equal toand being a full rank matrix such that

, , and. If the rank of is one,

then satisfies the constraints in the optimizationproblem of and trivially . Thus, we assumethat is greater than one and aim to show that based on ,another rank-one feasible point for the optimization problem of

can be constructed by following similar lines as in [35].To this end, let us consider the following set of equations

(43)

where an Hermitian matrix is an unknown variable.Due to the fact that , ,and are all real valued functions of ,the set of (43) is a linear set of 3 equations with variables,that is, the total number of real and imaginary variables in thematrix . Since the number of variables , is greaterthan the number of equations, there exists a nonzero solutiondenoted as for the linear set of equations (43). Let denotethe eigenvalue of the matrix which has the largest absolutevalue. Without loss of generality, we can assume that ,which is due to the fact that both and are solutions of

(43). Using and , we can construct a new matrix. It is then easy to verify that the expressions, , ,

and are valid and the rank of is less than orequal to . It is because the rank of the matrix isless than or equal to and the fact that rank of any matrixproduct is less than or equal to the rank of each of the matrices.It means that is another feasible point of the optimizationproblem of and its rank has reduced at least by one.This process can be repeated until or, equivalently, arank-one feasible point is found. After a rank one feasible point

is constructed, is also a feasiblepoint for the optimization problem of . Thus,which completes the proof.So far, we have shown that both optimal value functions

and have the same domain. Since the feasibleset of the optimization problem of is a subset of thefeasible set of the optimization problem of , we expectthat is less than or equal to at every feasiblepoint. However, due to the specific structure of the function

, these two optimal value functions are equivalent as itis shown in the following Theorem.Theorem 1: The optimal value functions and

are equivalent, i.e., , .Proof: In order to show that these optimal value functions

are equal, we use the dual problem of the optimization problemsof and . It is easy to verify that the optimizationproblems of and have the same following dualproblem

(44)

The following optimal value function can be defined based on(44)

(45)

Since the dual problem (44) gives an upper-bound for the op-timization problems of and , consequently, thefunction (45) is greater than or equal to andfor every . The optimization problem ofis convex and satisfies the Slater’s condition [34] because forevery , there exists a strictly feasible point for its dualproblem (44). Specifically, the point ,

, is a strictly feasible point for the dual problem(44) as it can be easily verified that the matrix

, or equivalently,is positive definite. Therefore, the

duality gap between the optimization problem of andits dual problem (44) is zero [34] which implies that for every

, .Regarding the optimization problem of which is a

QCQP problem [35]–[37], it is known that the duality gap be-tween a QCQP problem with three or less constraints and itsdual problem is zero [35]. Specifically, Corollary 3.3 of [35,Section 3] implies that the duality gap between the optimiza-tion problem of and its dual optimization problem (44)

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is zero and hence, . Since both of the optimiza-tion problems of and have zero duality gap withtheir dual problem (44), it can be concluded that in addition tohaving the same domain, the functions and havethe same optimal values, i.e., for every fea-sible .For any feasible point of the optimization problem of

denoted as , the matrix is also a feasible point ofthe optimization problem of and their corresponding ob-jective values are the same. Based on the latter fact and alsothe fact that the functions and have the sameoptimal values, it can be concluded that if denotes theoptimal solution of the optimization problem of , then

is also the optimal solution of the optimizationproblem of . Therefore, for every , there ex-ists a rank-one solution for the optimization problem of .The algorithm for constructing such a rank-one solution from ageneral-rank solution of the optimization problem of hasbeen explained in [35].Although the optimal value functions and

are equal, however, compared to the optimization problemof which is non-convex, the optimization problem of

is convex. Using this fact and replacing byin the original optimization problem (40), the problem

(36) can be simplified as

(46)

Therefore, instead of the original optimization problem (36),we can solve the simplified problem (46). Based on the optimalsolution of the simplified problem, denoted as , , and

, the optimal solution of the original problem can be found.The optimal values of and are equal to the corresponding op-timal values of the simplified problem, while the optimal canbe constructed based on using rank-reduction techniques[35] mentioned above.It is worth stressing that for every feasible point of the op-

timization problem (46) denoted as , the terms ,, and are positive, and therefore, the cor-

responding objective value is positive as well. The latter can beeasily verified by applying Lemma 1 of [38, Section 2] whichstates that for every Hermitian matrix and Hermitian positivesemi-definite matrix , is greater than or equal to

. Applying this lemma, it can be found that

(47)

where Lemma 1 of [38, Section 2] has been applied in thethird line of (47) and the last equality follows from the fact that

as is a feasible point. Since and arepositive definite, all the eigenvalues of the productare positive [31], and therefore, is positive. In

a similar way, it can be proved that andare necessarily positive, and therefore, the variables andare also positive. Thus, the task of maximizing the objectivefunction in the problem (46) is equivalent to maximizing thelogarithm of this objective function because is a strictlyincreasing function and the objective function in (46) is posi-tive. Then, the optimization problem (46) can be equivalentlyrewritten as

(48)

In summary, by replacing by , we are able towrite our optimization problem as a DC programming problem,where the fact that in the objective of (48) is aconcave function is also considered. Although the problem (48)boils down to the known family of DC programming problems,still there exists no solution for such DC programming prob-lems with guaranteed polynomial time complexity. However,the problem (48) has a very particular structure, such as, all theconstraints are convex and the terms andin the objective are concave. Thus, the only term that makes theproblem overall non-convex is the term in the objec-tive. If is piece-wise linearized over a finite number ofintervals,2 then the objective function becomes concave on theseintervals and the whole problem (48) becomes convex. The re-sulting convex problems over different linearization intervalsfor can be solved efficiently in polynomial time, andthen, the sub-optimal solution of the problem (48) can be found.The fact that such a solution is sub-optimal follows from the lin-earization, which has a finite accuracy. The smaller the intervalsare, the more accurate the solution of (48) becomes. However,such a solution procedure is not the most efficient in terms ofcomputational complexity. Thus, we develop another method(the POTDC algorithm) which makes it possible to find at leasta KKT solution for the problem (48) (with a great evidence thatsuch a solution is globally optimal) in a more efficient way.Let us introduce a new additional variable and then express

the problem (48) equivalently as

(49)

The objective function of the optimization problem (49) isconcave and all the constraints except the constraintare convex. Thus, we can develop an iterative method thatis different from the aforementioned piece-wise lineariza-tion-based method, and is based on linearizing the non-convexterm in the constraint around a suitably se-lected point in each iteration. More specifically, the linearizingpoint in each iteration is selected so that the objective functionincreases in every iteration of the iterative algorithm. Roughlyspeaking, the main idea of this iterative method is similar tothe gradient based Newton-type methods. In the first itera-tion, we start with an arbitrary point selected in the interval

and denoted as . Then2As explained before, the parameter can take values only in a finite interval.

Thus, a finite number of linearization intervals for is needed.

KHABBAZIBASMENJ et al.: SUM-RATE MAXIMIZATION IN TWO-WAY AF MIMO RELAYING 5485

the non-convex function can be replaced by its linearapproximation around this point , that is,

(50)

which results in the following convex optimization problem

(51)

The problem (51) can be efficiently solved using the interior-point-based numerical methods with the worst-case complexityof where is the total number of op-timization variables in the problem (51) including the real andimaginary parts of the elements of as well as , , and .Once the optimal solution of this problem, denoted in the firstiteration as , , , and , is found, the algorithmproceeds to the second iteration by replacing the functionby its linear approximation around found from the previous(first) iteration. Fig. 2 shows how is replaced by its linearapproximation around where is the optimal value ofobtained through solving (51) using such a linear approxima-tion. In the second iteration, the resulting optimization problemhas the same structure as the problem (51) in which has to beset to obtained from the first iteration. This process con-tinues, and th iteration is obtained by replacing by itslinearization of type (50) around found at the iteration

. The POTDC algorithm for solving the problem (49) issummarized in Algorithm 1.

Algorithm 1: The POTDC algorithm for solving theoptimization problem (49)

Initialize: Select an arbitrary from the interval, set the counter to be equal to 1 and

choose an accuracy parameter .

while The difference between the values of the objectivefunction in two consecutive iterations is larger than . do

Use the linearization of type (50) and solve theoptimization problem (51) to obtain , ,and .

Set , and .

.

end while

Output: .

The following two lemmas regarding the proposed POTDCalgorithm are of interest. First, the termination condition in thePOTDC algorithm is guaranteed to be satisfied due to the fol-lowing lemma which states that by choosing in the aboveproposed manner, the optimal values of the objective functionof (51) for , , and are non-decreasing.

Fig. 2. Linear approximation of around . The region above the dashedcurve is non-convex.

Lemma 2: The optimal values of the objective function ofthe optimization problem (51) obtained over the iterations ofthe POTDC algorithm are non-decreasing.

Proof: Considering the linearized problem (51) at the iter-ation . It is easy to verify that , , , andgive a feasible point for this problem. Therefore, the optimalvalue at the iteration must be greater than or equal to theoptimal value at the iteration which completes the proof.Second, the following lemma regarding the solution found by

the POTDC algorithm is of interest.Lemma 3: The solution obtained using the POTDC algorithm

satisfies the KKT conditions.Proof: This lemma follows straightforwardly from Propo-

sition 3.2 of [39, Section 3].As soon as the solution of the relaxed problem (49) is found,

the solution of the original problem (36), which is equivalentto the solution of the sum-rate maximization problem (35), canbe found using one of the existing methods for extracting arank-one solution [35]–[37]. The rank reduction-based tech-nique of [35] and the algebraic technique of [37] have beenmentioned earlier, while the method based on solving the dualproblem (see [36]) exploits the fact that a QCQP problem withonly two constraints has zero duality gap. Note that the compu-tational complexity of the POTDC algorithm is equivalent tothat of solving an SDP problem, i.e., , timesthe number of the iterations. Although the POTDC finds a KKTsolution for the considered sum-rate maximization problem, wealso aim at showing the evidence that such a solution is globallyoptimal. Toward this end, we will need an upper-bound for theoptimal value.

V. AN UPPER-BOUND FOR THE OPTIMAL VALUEThrough extensive simulations we have observed that regard-

less of the initial value chosen for in the first iteration ofthe POTDC algorithm, the proposed iterative method alwaysconverges to the global optimum of the problem (49). How-ever, since the original problem is not convex, this can not beeasily verified analytically. A comparison between the optimalvalue obtained by using the proposed iterative method and alsothe global optimal value can be, however, done by developing

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a tight upper-bound for the optimal value of the problem andcomparing the solution to such an upper-bound. Thus, in thissection, we find such an upper-bound for the optimal value ofthe optimization problem (49). For this goal, we first considerthe following lemma which gives an upper-bound for the op-timal value of the variable in the problem (49). This lemmawill further be used for obtaining the desired upper-bound forour problem.Lemma 4: The optimal value of the variable in (49)

or equivalently (48), denoted as , is upper-bounded by, where is the value of the objective function in the

problem (49) or equivalently (48) corresponding to any arbi-trary feasible point and is the optimal value of the followingconvex optimization problem3

(52)

Proof: Let , , and denote the optimal solutionof the optimization problem (48). We define another auxiliaryoptimization problem based on the problem (48) by fixing thevariable to as

(53)

For every feasible point of the problem (53), denoted as ,it is easy to verify that is also a feasible point of theproblem (48). Based on this fact, it can be concluded that theoptimal value of the problem (53) is less than or equal to the op-timal value of the problem (48). However, since is afeasible point of the problem (53) and the value of the objectivefunction at this feasible point is equal to the optimal value of theproblem (48), that is, ,we find that both optimization problems (53) and (48) have thesame optimal value. To find an upper-bound for , we makethe feasible set of the problem (53) independent of whichcan be done by dropping the constraint inthe problem (53). Then the following problem is obtained:

(54)

Noticing that the feasible set of the optimization problem (53)is a subset of the feasible set of the problem (54), it is straight-forward to conclude that the optimal value of the problem (54)is equal to or greater than the optimal value of the problem(53) and, thus, also the optimal value of the problem (48). Onthe other hand, is the value of the objective function of theproblem (48) which corresponds to an arbitrary feasible pointand as a result is less than or equal to the optimal value of theproblem (48). Since the optimal value of the problem (54) isgreater than or equal to the optimal value of the problem (48)and the optimal value of the problem (48) is greater than orequal to , the optimal value of the problem (54), denotedas , is greater than or equal to , and therefore,

which completes the proof.3Note that this optimization problem can be solved efficiently using numerical

methods, for example, interior point methods.

Fig. 3. Feasible region of the constraint and the convex hull in eachsub-division.

Note that as mentioned earlier, is the objective value ofthe problem (48) that corresponds to an arbitrary feasible point.In order to obtain the tightest possible upper-bound for ,we choose to be the largest possible value that we alreadyknow. A suitable choice for is then the one which is ob-tained using the POTDC algorithm. In other words, we chooseas the corresponding objective value of the problem (48)

at the optimal point which is resulted from the POTDC algo-rithm. Thus, we have obtained an upper-bound for whichmakes it further possible to develop an upper-bound for the op-timal value of the optimization problem (48). To this end, weconsider the only non-convex constraint of this problem, i.e.,

. Fig. 3 illustrates a subset of the feasible regioncorresponding to the non-convex constraint where

equals , i.e., the smallest value of forwhich the problem (49) is feasible, and is the upper-boundfor the optimal value given by Lemma 4 ( is equalto if it is smaller than the upper-bound of

obtained using Lemma 4). For obtaining an upper-boundfor the optimal value of the problem (49), we divide the in-terval into sections as it is shown in Fig. 3.Then, each section is considered separately. In each such sec-tion, the corresponding non-convex feasible set is replaced byits convex-hull and each corresponding optimization problem issolved separately as well. The maximum optimal value of suchconvex optimization problems is then the upper-bound. In-

deed, solving the resulting convex optimization problems andchoosing the maximum optimum value among them is equiva-lent to replacing the constraint with the feasible setwhich is described by the region above the solid line in Fig. 3.The upper-bound becomes more and more accurate when thenumber of the intervals, i.e., increases.

VI. SEMI-ALGEBRAIC SOLUTION VIA GENERALIZEDEIGENVECTORS (RAGES)

In this section we present RAGES as an alternative solutionto the sum-rate maximization problem (35) which is based ongeneralized eigenvectors. It requires a different parameteriza-tion than the one used in the POTDC algorithm and in somecases it is more computationally efficient.

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A. Basic Approach: Generalized EigenvectorsTo derive the link between (35) and generalized eigenvec-

tors we start with the necessary condition for optimality that thegradient of (35) vanishes. Therefore, if we find all vectors forwhich the gradient of the objective functions is zero, the globaloptimum must be one of them. By using the product rule andthe chain rule of differentiation, the condition of zero gradientcan be expressed as [27]

(55)

Rearranging (55) we obtain

(56)

where and are defined via

and (57)

It follows from (56) that the optimal must be a general-ized eigenvector of the pair of matrices and

. Moreover, the corresponding generalizedeigenvalue is given by which is logarithmically propor-tional to the rate of the terminal one . Unfortunately, the ma-trices and contain the parameters

and which also depend on and are hence not known inadvance. Therefore, we still need to optimize over these two pa-rameters. However, compared to the original problem of findinga complex-valued matrix, optimizing over the tworeal-valued scalar parameters is significantly simpler. The fol-lowing subsections show how to simplify this 2-D search evenfurther.

B. Bounds on the Parameters andSince both parameters and have a physical interpre-

tation, the lower and upper-bounds for them can be easily found.Such bounds are useful since they limit the search space that hasto be tested. For instance, can be expanded into

(58)

The quadratic forms can be bounded by using the fact that forany Hermitian matrix we have

(59)

where and are the smallest and the largesteigenvalues of , respectively, and . It follows from(27) that

and (60)

where is a short hand notation for . Furthermore, ingeneral the following inequality holds

(61)

which for the case of white noise at the relay boils down to thefollowing tighter condition .

The relations (60) and (61) can be used to bound (58). Specif-ically, an upper-bound for can be found by upper-boundingthe enumerator and lower-bounding the denominator, while thelower-bound can be found by lower-bounding the enumeratorand upper-bounding the denominator. This yields

(62)

(63)

where and can be dropped if the noise at the relay iswhite. However, an upper-bound for is till needed. Dueto the relay power constraint we have .Using the latter condition, the following bound can be de-rived . However, it is easy to check that thisbound is very loose since for white noise at the relay we have

and for arbitrary relay noise covariancematrices no lower-bound exists (the infimum over is zero).This bound is so loose because it is extremely pessimistic: itmeasures the norm of in the case when only noise is amplifiedand no power is put on the eigenvalues related to the signals ofinterest. However, such a case is practically irrelevant since itcorresponds to a sum-rate equal to zero. Therefore, we proposeto replace in (62) and (63) by4

(64)

where is the second largest eigenvalue of the matrix.In a similar manner, can be bounded. In this case, the enu-

merator and the denominator have the additional termsand , respectively. A pes-

simistic (loose) bound is obtained by bounding these two termsindependently, i.e.,and . This yields

(65)

(66)

Again, these bounds are pessimistic since they assume that thereexists an optimal relay strategy for which

but , i.e., the rate of the second terminal isequal to zero. However, it is typically sum-rate optimal to havesignificantly more balanced rates between the two users. In fact,for the “symmetric” scenario when , ,

, 2, and , we always have atoptimality. Therefore, these bounds can be further tightened ifa priori knowledge about the specific scenario is available.

C. Efficient 2-D and 1-D SearchOnce the search space for and has been fixed, we

can find the maximum via optimization over these two param-4We have observed in all our simulations that this value poses indeed an

upper-bound on the norm of the optimal solution .

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Fig. 4. Sum-rate versus and for ,, .

eters using a 2-D search. In general, a 2-D exhaustive searchcan be computationally demanding, i.e., the computational com-plexity will be higher than that of the POTDC algorithm. How-ever, as we show in the sequel, for the problem at hand, thissearch can be implemented efficiently. These efficient imple-mentations are, however, heuristic since they rely on propertiesof the objective functions that are apparent by visual inspection.As we will see in simulations, the performance of the resultingRAGES algorithm coincides with that of the rigorous POTDCalgorithm.Fig. 4 demonstrates a typical example of the sum-rate

as a function of and . For this example we have chosen, ,

and we have drawn the channel vectors from an uncorre-lated Rayleigh fading distribution assuming reciprocity. By vi-sual inspection, this sample objective function shows two in-teresting properties. First, it is a quasi-convex function with re-spect to the parameters and which allows for efficient(quasi-convex) optimization tools for finding its maximum. Al-beit this property is only demonstrated for one example here,it has been always present in our numerical evaluations evenwhen largely varying all system parameters. Second, for everyvalue of the corresponding maximization over yieldsone maximal value which depends on only very weakly.This is illustrated by Fig. 5 which displays the relative changeof the objective function for different choices of ,each time optimizing it over . The displayed values representthe relative decrease of the objective functions compared to theglobal optimum, i.e., for the worst choice of , the achievedsum-rate is about lower than for the bestchoice of . Consequently, the 2-D search over andcan be replaced essentially without any loss by a 1-D searchover only for one fixed value of (e.g., the geometricmean of the upper and the lower-bound).In addition, instead of performing the search directly over the

original objective function , we can find an even simplerobjective functions by using the physical meaning of our twosearch parameters. To this end, let us introduce a new parameter

as a function of as follows

(67)

Fig. 5. Relative change in sum-rate versus : optimizing overfor every choice of . The same data set as in Fig. 4 is used.

Fig. 6. Objective function . The same data set as in Fig. 4 isused. The dashed line indicates the points where .

Here is the relay weight vector at the current search point. Then we know that in the optimal point , we

have . This can be used to construct a newobjective function

(68)

where is the dominant eigenvector of (56) for the currentsearch point ( ).Using the same data set as before, we display the corre-

sponding shape of in Fig. 6. The dashed lineindicates the set of points where . It canbe observed that for every value of , is amonotonic function in . Therefore, the bisection methodcan be used to find a zero crossing in which coincides withthe sum-rate-optimal for a given .

D. SummaryIn summary, it can be concluded that the RAGES approach

simplifies the optimization over a complex-valuedmatrix into the optimization over two real-valued parameterswhich both have a physical interpretation. Even more, the 2-Dsearch can be simplified into a 1-D search by fixing one of theparameters. The loss incurred to this step is typically small. In

KHABBAZIBASMENJ et al.: SUM-RATE MAXIMIZATION IN TWO-WAY AF MIMO RELAYING 5489

the example provided above, it is only 0.002%, but even varyingthe system parameters largely and using many random trials wenever found a relative difference higher than a few percents.Moreover, the 1-D search can be efficiently implemented by

exploiting the quasi-convexity of or the monotonicity of(e.g., using the bisection method). Again, these properties

are only demonstrated by examples but we have observed inall our simulations that the resulting algorithm yields a sum-rate very close to the optimum found by the exact solution andits upper-bound described before. This comparison is furtherillustrated in next section via numerical simulations.Comparing the POTDC and RAGES approaches, it is notice-

able that the POTDC approach is rigorous, while the RAGESapproach is at some points heuristic. As it has been mentioned,the computational complexity of solving the proposed sum-ratemaximization problem for two-way AF MIMO relaying usingthe POTDC algorithm is the same as the complexity of solvingthe SDP problem (49) product the number of iterations, wherethe number of iterations is significantly smaller than the di-mension of the problem. For example, for a relay with 4 an-tennas (the corresponding dimension of the problem is 16), thenumber of iterations is only 4–6. Alternatively, the complexityof solving the same problem using the RAGES approach isequivalent to the complexity of finding the dominant general-ized eigenvector, which has to be performed for each combina-tion of the parameters and . Since the search over oneparameter only is sufficient, the complexity of the RAGES ap-proach is typically lower than that of the POTDC algorithm forthe 1-D RAGES.

VII. SIMULATION RESULTS

In this section, we evaluate the performance of the new pro-posed methods via numerical simulations. Consider a commu-nication system consisting of two single-antenna terminals andan AF MIMO relay with antenna elements. The communi-cation between the terminals is bidirectional, i.e., it is performedbased on the two-way relaying scheme. It is assumed that per-fect channel knowledge is available at the terminals and at therelay, while the terminals use only effective channels (scalars),but the relay needs full channel vectors. The relay estimates thecorresponding channel coefficients between the relay antennaelements and the terminals based on the pilots which are trans-mitted from the terminals. Then based on these channel vectors,the relay computes the relay amplification matrix and thenuses it for forwarding the pilot signals to the terminals. After re-ceiving the forwarded pilot signals from the relay via the effec-tive channels, the terminals can estimate the effective channelsusing a suitable pilot-based channel estimation scheme, e.g., theLS.The noise powers of the relay antenna elements and the

single-antenna terminals , , and are assumedto be equal to . Uncorrelated Rayleigh fading channelsare considered and it is assumed that reciprocity holds, i.e.,

for 1, 2. The relay is assumed to be locatedover a line of unit length which connects the terminals to eachother and the variances of the channel coefficients betweenterminal and the relay antenna elements are all assumed to beproportional to , where is the distance betweenthe relay and the terminal and is the path-loss exponent

Fig. 7. Sum-rate versus for antennas. Example 1: thecase of symmetric channel conditions.

which is assumed to be equal to 3 throughout the simulations.5For obtaining each point, 100 independent simulation runs areused unless otherwise is specified.In order to design the relay amplification matrix , five

different methods are considered including the proposedPOTDC, 2-D RAGES and 1-D RAGES algorithms, the alge-braic norm-maximizing (ANOMAX) transmit strategy of [40]and the discrete Fourier transform (DFT) method that choosesthe relay precoding matrix as a scaled DFT matrix. Note thatthe ANOMAX strategy provides a closed-form solution for theproblem. Also note that for the DFT method no channel knowl-edge is needed. Thus, the DFT method serves as a benchmarkfor evaluating the gain achieved by using channel knowledge.The upper-bound is also shown in simulation examples 1 and2. For obtaining the upper-bound, the interval isdivided in 30 segments. In addition, the proposed techniquesare compared to the SNR-balancing technique of [41] for thescenario when multiple single-antenna relay nodes are usedand the method of [41] is applicable.

A. Example 1: Symmetric Channel Conditions

In our first example, we consider the case when the channelsbetween the relay antenna elements and both terminals have thesame channel quality. More specifically, it is assumed that therelay is located in the middle of the connecting line between theterminals and the transmit powers and and the totaltransmit power of the MIMO relay are all assumed to beequal to 1.Fig. 7 shows the sum-rate achieved by different aforemen-

tioned methods versus for the case of . It can beseen in this figure that the performance of the proposed methodscoincides with the upper-bound. Thus, the methods performoptimally in terms of providing the maximum sum-rate. TheANOMAX technique performs close to the optimal, while theDFT method gives a significantly lower sum-rate.5It is experimentally found that typically (see [42, p. 46–48] and

references therein). However, can be smaller than 2 when we have a wave-guide effect, i.e., indoors in corridors or in urban scenarios with narrow streetcanyons.

5490 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 10, OCTOBER 2012

Fig. 8. Sum-rate versus the distance between the relay and the secondterminal for antennas. Example 2: the case of asymmetric channelconditions.

B. Example 2: Asymmetric Channel Conditions

In the second example, we consider the case when the chan-nels between the relay antenna elements and the second terminalhave better channel quality than the channels between the relayantenna elements and the first terminal. Thus, we evaluate theeffect of the relay location on the achievable sum-rate. Particu-larly, we consider the case when the distance between the relayand the second terminal is less than or equal to the distancebetween the relay and the first terminal . The total transmitpower of the terminals, i.e., and and the total transmitpower of the MIMO relay all are assumed to be equal to1 and the noise powers in the relay antenna elements and theterminals all are assumed to be equal to 1.Fig. 8 shows the sum-rate achieved in this scenario by dif-

ferent methods tested versus the distance between the relay andthe second terminal , for the case of . It can be seen inthis figure that the proposed methods perform optimally, whilethe performance (sum-rate) of ANOMAX is slightly worse.As mentioned earlier, it is guaranteed that the POTDC al-

gorithm converges to at least a KKT solution of the sum-ratemaximization problem. However, our extensive simulation re-sults confirm that the POTDC algorithm converges to the globalmaximum of the problem in all simulation runs. It is approvedby the fact that the performance of the POTDC algorithm coin-cides with the upper-bound.Moreover, the 2-D RAGES and 1-DRAGES are, in fact, globally optimal, too. The ANOMAX andDFT methods, however, do not achieve the maximum sum-rate.The loss in sum-rate related to the DFT method is quite signifi-cant while the loss in sum-rate related to the ANOMAXmethodgrows from small in the case of symmetric channel conditionsto significant in the case of asymmetric channel conditions. Al-though ANOMAX enjoys a closed-form solution and it is evenapplicable in the case when terminals have multiple antennas,it is not a good substitute for the proposed methods because ofthe significant gap in performance in the asymmetric case.

C. Example 3: Effect of the Number of Relay Antenna Elements

In this example, we consider the effect of the number of relayantenna elements on the achievable sum-rate for the afore-mentioned methods. The powers assigned to the first and second

Fig. 9. Sum-rate versus the number of the relay antenna elements .Example 3: the case of asymmetric channel conditions.

terminals as well as to the relay are all equal to 1. The relay is as-sumed to be located at the distance of 1/5 from the second user.Moreover, the noise powers at the terminals and at the relay an-tenna elements are all assumed to be equal to 1. For obtainingeach point in this simulation example, 200 independent runs areused.Fig. 9 depicts the sum-rates achieved by different methods

versus the number of relay antenna elements . As it is ex-pected, by increasing (thus, increasing the number of de-grees of freedom), the sum-rate increases. For the DFT method,the sum-rate does not increase with the increase of becauseof the lack of channel knowledge for this method. The proposedmethods achieve higher sum-rate compared to ANOMAX.

D. Example 4: Performance Comparison for the Scenario ofTwo-Way Relaying via Multiple Single-Antenna RelaysIn our last example, we compare the proposed methods with

the SNR balancing-based approach of [41]. The method of [41]is developed for two-way relaying systems which consist oftwo single-antenna terminals and multiple single-antenna relaynodes. Subject to the constraint on the total transmit power ofthe relay nodes and the terminals, the method of [41] designsa beamforming vector for the relay nodes and the transmitpowers of the terminals to maximize the minimum receivedSNR at the terminals. In order to make a fair comparison,we consider a diagonal structure for the relay amplificationmatrix that corresponds to the case of [41] when multiplesingle-antenna nodes are used for relaying. It is worth men-tioning that for imposing such a diagonal structure for the relayamplification matrix in POTDC and RAGES, the vector

is replaced withand the matrices and , , 2 are replaced with newsquare matrices and , , 2 of size suchthat and

, . Moreover,for the proposed methods, we assume fixed transmit powersat the terminals and fixed total transmit power at the relaynodes that are all equal to 1, while for the method of [41], thetotal transmit power at the relay nodes and the terminals isassumed to be equal to 3. Thus, the overall consumed powerby the proposed methods and the method of [41] is the same,

KHABBAZIBASMENJ et al.: SUM-RATE MAXIMIZATION IN TWO-WAY AF MIMO RELAYING 5491

Fig. 10. Sum-rate versus for antennas. Example 4: thecase of symmetric channel conditions and distributed single antenna relays.

however, compared to [41], which also optimizes the powerusage of the terminals, the transmit powers of the terminalsin the proposed methods are fixed. The relay is assumed tolie in the middle in between the terminals. Fig. 10 shows thecorresponding performance of the methods tested. From thisfigure it can be seen that the proposed methods demonstrate abetter performance compared to the method of [41] as it maybe expected even though they use a fixed transmit power forthe terminals.

VIII. CONCLUSIONS AND DISCUSSIONSWe have shown that the sum-rate maximization problem in

two-way AF MIMO relaying belongs to the class of DC pro-gramming problems. Although DC programming problems canbe solved by the branch-and-bound method, this method doesnot have any polynomial time guarantees for its worst-case com-plexity. In this paper, we have developed the so-called POTDCalgorithm for finding a KKT solution of the aforementionedproblem with polynomial time worst-case complexity. There is,however, a great evidence that the global optimal solution is alsoachieved. The other method called RAGES shows the same per-formance as the POTDC algorithm. The POTDC algorithm isbased on a specific parameterization of the objective function,that is, the product of quadratic fractional functions; and thenapplication of SDP relaxation, linearization and iterative searchover a single parameter. Its design is rigorous and is based on therecent advances in convex optimization. The RAGES algorithmis based on a different parameterization of the objective func-tion and the generalized eigenvectors method. It may enjoy alower computational complexity that makes it a valid alternativeto the POTDC algorithm especially if 1-D search is used. Theupper-bound for the solution of the problem is developed andit is demonstrated by simulations that both proposed methodsachieve the upper-bound and are, thus, globally optimal.The proposed POTDC algorithm represents a general opti-

mization technique applicable for solving a wide class of DCprogramming problems. Essentially, the optimization problemsconsisting of the maximization/minimization of a product ofquadratic fractional functions can be handled using the pro-posed POTDC approach. Moreover, the POTDC algorithm canbe used for solving optimization problems with constraints rep-resented as a difference of two quadratic forms. More general

problems can be also addressed by POTDC approach by ap-plying some relatively straightforward modifications. For ex-ample, if the problem is to optimize a product of more than twoquadratic fractional functions under a single quadratic (power)constraint, the number of constraints in the corresponding DCprogramming problem will be more than three. In this case, ran-domization procedures have to be adopted to recover a rank-onesolution from the solution of the relaxed problem. Such solu-tions obviously may not be exact.Other signal processing problems that can be addressed using

the proposed POTDC approach are the general-rank robustadaptive beamformer with a positive semi-definite constraint,the dynamic spectrum management for digital subscriber lines,the problems of finding the weighted sum-rate point, the pro-portional-fairness operating point, and the max-min optimalpoint for the MISO interference channel, the problem of robustbeamforming design for AF relay networks with multiple relaynodes, the proportional fairness as well as the max-min rateproblem for two-way relaying, and so on. The extensions of thePOTDC approach to some of the aforementioned problem is anissue of future research.

APPENDIX AThe matrix is rank deficient only if the noise is spatially

correlated. Let the rank of be denoted as , .Since the matrix , its rank is equal to the rankof the matrix multiplied by the rank of the matrix , i.e.,

. Then if is rank deficient, is alsorank deficient.For convenience, we restate the sum-rate maximization

problem (29) as(69)

where

(70)

If is rank deficient, then for every whereis the null space of , the total transmit power from

the relay is zero, i.e., . Moreover, the corre-sponding sum-rate for any vector is equal tozero. To show this, let us considerthat straightforwardly implies that

(71)

Note that the first equality in (71) follows from (3). Substituting(4) in (71), we obtain

(72)

Since the matrices ,

, and in (72) are all

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positive semi-definite and the powers and are strictlypositive, (72) is satisfied only if , ,and . Therefore, the following equations are inorder

(73)

(74)

(75)Substituting (73)–(75) in (70), we conclude that the sum-rate isindeed zero for any . Furthermore, in a similarway, it can be shown that, for any such that

, , and, which means that does not have any contribution

in the transmit power as well as the sum-rate.Using the above observations, it is easy to see that if is

rank deficient, the only thing required to do is reformulating therate function (70) in the following manner. Denote the eigen-value decomposition of the matrix aswhere and are unitary and diagonal ma-trices of eigenvectors and eigenvalues, respectively. The theigenvector and the th eigenvalue of denoted as and ,respectively, constitutes the column of and th diagonal el-ement of . It is assumed without loss of generality that theeigenvalues , are ordered in the descendingorder, i.e., , . Since in the case ofrank deficient , the rank of is equal to , the last

eigenvalues of are zero. By splitting tothe matrix and thematrix as , the matrix can be decomposedas where thediagonal matrix contains the dominant eigenvalues,while the other diagonalmatrix contains the zero eigenvalues. Sinceis unitary, any arbitrary vector can be expressed as

where and are the coef-ficient vectors. It is easy to verify that the component liesinside and as a resultand

. Therefore, can be any arbitrary vector and weonly need to find the optimal . Substituting in (70), thesum-rate can be expressed only as a function of as follows

(76)

Then the optimization problem (69) is equivalent to the maxi-mization of (76) under the constraint

(77)

Since the matrix is full rank, the corresponding optimizationproblem can be solved by the methods that we develop in thepaper.

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Arash Khabbazibasmenj (S’08) received the B.Sc.degree from Amirkabir University of Technology,Tehran, Iran, in 2006 and the M.Sc. degree inelectrical engineering from the University of Tehran,Tehran, Iran, in 2009.He is currently working toward the Ph.D. degree

in electrical engineering at the University of Alberta,Edmonton, AB, Canada. During spring and summer2011, he was also a visiting student at IlmenauUniversity of Technology, Germany. His researchinterests include signal processing and optimization

methods in radar, communications and related fields. Mr. Khabbazibasmenjreceived the Alberta Innovates graduate award in ICT.

Florian Roemer (S’04) has studied computerengineering at Ilmenau University of Technology,Germany, and McMaster University, Canada. Hereceived the Diplom-Ingenieur (M.S.) degree incommunications engineering from Ilmenau Univer-sity of Technology in October 2006. He received theSiemens Communications Academic Award in 2006for his diploma thesis.Since December 2006, he has been a Research

Assistant in the Communications Research Labora-tory at the Ilmenau University of Technology. His

research interests include multidimensional signal processing, high-resolutionparameter estimation, as well as multi-user MIMO precoding and relaying.

Sergiy A. Vorobyov (M’02–SM’05) received theM.Sc. and Ph.D. degrees in systems and control fromKharkiv National University of Radio Electronics,Ukraine, in 1994 and 1997, respectively.Since 2006, he has been with the Department of

Electrical and Computer Engineering, University ofAlberta, Edmonton, AB, Canada, where he becomean Associate Professor in 2010 and Full Professorin 2012. Since his graduation, he also held variousresearch and faculty positions at Kharkiv NationalUniversity of Radio Electronics, Ukraine; the Insti-

tute of Physical and Chemical Research (RIKEN), Japan; McMaster University,Canada; Duisburg-Essen University and Darmstadt University of Technology,Germany; and the Joint Research Institute between Heriot-Watt University andEdinburgh University, U.K. He has also held visiting positions at Technion,Haifa, Israel, in 2005 and Ilmenau University of Technology, Ilmenau, Ger-many, in 2011. His research interests include statistical and array signal pro-cessing, applications of linear algebra, optimization, and game theory methodsin signal processing and communications, estimation, detection, and samplingtheories, and cognitive systems.Dr. Vorobyov is a recipient of the 2004 IEEE Signal Processing Society Best

Paper Award, the 2007 Alberta Ingenuity New Faculty Award, the 2011 CarlZeiss Award (Germany), and other awards. He was an Associate Editor for theIEEE TRANSACTIONS ON SIGNAL PROCESSING from 2006 to 2010 and for theIEEE SIGNAL PROCESSING LETTERS from 2007 to 2009. He is a member ofthe Sensor Array and Multi-Channel Signal Processing and Signal Processingfor Communications and Networking Technical Committees of the IEEE SignalProcessing Society. He has served as the Track Chair for Asilomar 2011, PacificGrove, CA, the Technical Co-Chair for IEEE CAMSAP 2011, Puerto Rico, andthe Plenary Chair of ISWCS 2013, Ilmenau, Germany.

Martin Haardt (S’90–M’98–SM’99) studiedelectrical engineering at the Ruhr UniversityBochum, and at Purdue University. He received theDiplom-Ingenieur (M.S.) degree from the Ruhr-Uni-versity Bochum in 1991 and the Doktor-Ingenieur(Ph.D.) degree from Munich University of Tech-nology, Germany, in 1996.In 1997, he joined Siemens Mobile Networks,

Munich, Germany, where he was responsible forstrategic research for third-generation mobile radiosystems. From 1998 to 2001, he was Director for

International Projects and University Cooperations in the mobile infrastructurebusiness of Siemens, where his work focused on mobile communicationsbeyond the third generation. While at Siemens, he also taught in the interna-tional Master of Science in Communications Engineering program at MunichUniversity of Technology. Since 2001, he has been a Full Professor in theDepartment of Electrical Engineering and Information Technology and Headof the Communications Research Lab at Ilmenau University of Technology,Germany. Since 2012, he has also served as an Honorary Visiting Professorin the Electronics Department at the University of York, U.K. In fall 2006and 2007, he was a Visiting Professor at University of Nice, Sophia-Antipolis,France, and at University of York, U.K., respectively. His research interestsinclude wireless communications, array signal processing, high-resolutionparameter estimation, and numerical linear and multi-linear algebra.Dr. Haardt has received the 2009 Best Paper Award from the IEEE Signal

Processing Society, the Vodafone (formerly Mannesmann Mobilfunk) Innova-tions-Award for outstanding research in mobile communications, the ITG BestPaper Award from the Association of Electrical Engineering, Electronics, andInformation Technology (VDE), and the Rohde & Schwarz Outstanding Disser-tation Award. He has served as an Associate Editor for the IEEE TRANSACTIONSON SIGNAL PROCESSING from 2002 to 2006 and since 2011, the IEEE SIGNALPROCESSING LETTERS from 2006 to 2010, the Research Letters in Signal Pro-cessing from 2007 to 2009, the Hindawi Journal of Electrical and ComputerEngineering since 2009, the EURASIP Signal Processing Journal since 2011,and as a Guest Editor for the EURASIP Journal on Wireless Communicationsand Networking. He has also served as an elected member of the Sensor Arrayand Multichannel (SAM) Technical Committee of the IEEE Signal ProcessingSociety since 2011, as the Technical Co-Chair of the IEEE International Sym-posium on Personal Indoor and Mobile Radio Communications (PIMRC) 2005in Berlin, Germany, as the Technical Program Chair of the IEEE InternationalSymposium onWireless Communication Systems (ISWCS) 2010 in York, U.K.,and as the General Chair of ISWCS 2013 in Ilmenau, Germany.